Medical Image Reconstruction by Gengsheng Lawrence Zeng

Author:Gengsheng Lawrence Zeng
Language: eng
Format: epub
Publisher: De Gruyter
Published: 2023-05-12T06:43:05.904000+00:00

6.6.2

Iterative methods: stopping early

There are many ways to control the noise in an iterative algorithm. We can first consider a rough noise propagation model of a linear iterative algorithm:

(6.30)

where Errorimage is the error magnitude in the reconstructed image, Errorprojections is the error magnitude in the projections, and n(ω) is the algorithm transfer function which depends on the frequency ω and the iteration number n.

We can compare an iterative reconstruction algorithm with an SVD matrix pseudo-inverse solution. You may imagine that n(ω) contains the information of both the singular values and singular vectors of the imaging matrix A. The frequency components are in the singular vectors. As the iteration number n increases, more singular vectors join n(ω). The iteration number is somehow related to the cutoff index in an SVD pseudo-inversion expression. With a larger iteration number n, n(ω) contains components with higher frequencies. In some linear algorithms, this relationship can be simplified to

(6.31)

where κ is similar to the condition number of matrix A, and κ is defined as the ratio of the largest singular value σ1 over the cutoff singular value σn. This simplification is reasonable because the “worst” noise influence comes from the frequency components (i.e., the singular vector) corresponding to the current smallest singular value σn. In the SVD pseudo-inverse method, the reconstructed image is a sum of many terms. Each term is a product of a component (i.e., the singular vector) and the reciprocal of its corresponding singular value 1/σ. The largest gain comes from 1/σn, which corresponds to a singular vector with very high frequencies.

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.